An Introduction to Langlands Functoriality

Abstract

The Principle of Functoriality has long been regarded as the centre of the Langlands Program. More recently, it has had to share the spotlight with Reciprocity, Langlands’ conjecture that relates automorphic representations with motives from algebraic geometry. However, the two principles are closely related, and in any case, Reciprocity came at the end of the decade that followed the years 1960–1967 that are the focus of this volume. Functoriality famously had its roots in the seventeen-page letter that Langlands gave to André Weil in 1967 [L2]. He wrote some of the details shortly afterwards in the article he dedicated to Salomon Bochner [L3]. It represented a very different direction for Langlands after his monumental volume [L1] on Eisenstein series, which was largely analysis. Langlands credits Bochner, an analyst himself, with directing him towards number theory, especially, I believe, class field theory and its long-sought nonabelian generalization. There are several ways to introduce functoriality to a general reader. One is as a series of identities (reciprocity laws) that relate families of conjugacy classes c = {cp : p N}